Brjuno number

In mathematics, a Brjuno number is an irrational number α such that

$\sum _{n=0}^{\infty }{\frac {\log q_{n+1}}{q_{n}}}<\infty$

where p_n/q_n are the convergents of the continued fraction expansion of α. They were introduced by Brjuno (1971), who showed that germs of holomorphic functions with linear part e^2πiα are linearizable if α is a Brjuno number. Yoccoz (1995) showed that this condition is also necessary for quadratic polynomials. For other germs the question is still open.

Brjuno function

The real Brjuno function B(x) is defined for irrational x and satisfies

$B(x)=B(x+1)$

$B(x)=-\log x+xB(1/x)$ for all irrational x between 0 and 1.

References

• Brjuno, A. D. (1971), "Analytic form of differential equations. I, II", Trudy Moskovskogo Matematičeskogo Obščestva, 25: 119–262, ISSN 0134-8663, MR 0377192
• Marmi, Stefano; Moussa, Pierre; Yoccoz, Jean-Christophe (2001), "Complex Brjuno functions", Journal of the American Mathematical Society, 14 (4): 783–841, doi:10.1090/S0894-0347-01-00371-X, ISSN 0894-0347, MR 1839917
• Yoccoz, Jean-Christophe (1995), "Petits diviseurs en dimension 1", Astérisque, 231: 3–88, ISSN 0303-1179, MR 1367353 {{citation}}: |chapter= ignored (help)